Q63PE - College Physics (Urone)

Question

(a) What is the energy stored in the $\mathbf{10}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{\mu F}$ capacitor of a heart defibrillator charged to $\mathbf{9}\mathbf{.}\mathbf{00}\mathbf{\times }{\mathbf{10}}^{\mathbf{3}}\mathbf{ }\mathbf{V}$? (b) Find the amount of stored charge.

Step-by-Step Solution

Answer

1. The stored energy is 405 J.
2. The charge stored is 0.09 C

Step 1: Given Data

The capacitance of the capacitor is:

 $\begin{array}{rcl}\mathrm{C}& =& (10.0\mathrm{\mu F})\left(\frac{1\mathrm{F}}{{10}^6\mathrm{\mu F}}\right)\\ & =& 10.0\times {10}^{-6}\mathrm{F}\\ & & \end{array}$

     The voltage between the capacitor plates is: $\mathrm{\Delta V}=9.00\times {10}^3\mathrm{V}$

 

Step 2: Calculating for part (a) energy of capacitor

The stored energy for a capacitor of capacitance with a potential difference between the plates is given by:

 ${\mathbf{U}}_{\mathbf{E}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{C}{\left(\mathrm{\Delta V}\right)}^{\mathbf{2}}$      

 a)

    The capacitance C is defined as charge stored per unit potential difference

     $\mathbf{C}\mathbf{=}\frac{\mathbf{Q}}{\left(\mathrm{\Delta V}\right)}$

    Thus, entering the values of C and $\left(\Delta V\right)$, we obtain:

   $\begin{array}{rcl}{\mathbf{U}}_{\mathbf{E}}& \mathbf{=}& \frac{\mathbf{1}}{\mathbf{2}}\left(10.0\times {10}^{-6}F\right)\left(9.00\times {10}^{3}V\right)\\ & \mathbf{=}& \mathbf{405}\mathbf{J}\\ & & \end{array}$

   Therefore, stored energy is  ${\mathit{U}}_{\mathbf{E}}\mathbf{=}\mathbf{\text{ 405 J}}$

Step 3: Calculating for part (b), stored charge

b)

  The stored charge is given as

 $\mathit{Q}\mathbf{=}\mathit{C}\mathbf{\times }\mathit{\Delta }\mathit{V}$

  Substitute numerical values:

  $$\begin{gathered} \mathbf{\text{Q=}}\left(10.0\times {10}^{-6} F\right)\left(9.00\times {10}^3 V\right) \\ \mathbf{=}\mathbf{0}\mathbf{.09}\mathbf{C} \end{gathered}$$

Therefore, the charge accumulated on capacitor plates is$\mathit{Q}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{09}\mathbf{ }\mathbf{\text{C}}$